#7969691

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2,3,4)(6,9)(10,11)(15,21)(18,25)(22,27)(26,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,20)(15,18,21,25)(16,19,24,23)(22,26,27,28);(5,6,8,9)(7,10,12,11)(13,14)(15,18)(16,23)(17,20)(19,24)(21,25)(22,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27);(5,6)(7,11)(8,9)(10,12)(13,16)(14,19)(15,22)(17,24)(18,26)(20,23)(21,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,23)(17,20)(19,24)(21,25)(22,28)(26,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308335, #72308489, #72308503
maximal normal subgroup of	#72308335, #72308489, #72308503

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7876887);